Hedging-Pricing Duality

Prof. Halil Soner

ETH Zurich

About the Speaker

Soner is currently a Professor at the Swiss Federal Institute of Technology in Zurich, Eidgenössische Technische Hochschule Zürich (ETH-Z) and also holds a senior chair at the Swiss Finance Institute. His research is on nonlinear analysis with emphasis on optimal stochastic control, partial differential equations, stochastic processes and mathematical finance.

Prior to moving to Zurich, he has spent nine years in Istanbul, Turkey and nineteen years in the United States of America. During his tenure in Turkey, he held the Isik Inselbag Chair at Sabanci University for two years and was a member of the Mathematics Department at Koc University for seven years prior to that. He has received his doctoral degree from the Division of Applied Mathematics of Brown University. Later, he was a member of the Department of Mathematical Sciences at Carnegie Mellon 1986 - 1998. In 1998, he became the Paul M. Wyhtes '55 Professor of Engineering and Finance at Princeton.

He has co-authored a book, with Wendell Fleming, on viscosity solutions and stochastic control; Controlled Markov Processes and Viscosity Solutions, and authored or co-authored several articles on nonlinear partial differential equations, viscosity solutions, stochastic optimal control and mathematical finance. Since 2011, he has been the Executive Secretary of the Bachelier Finance Society and in 2014 he was awarded the Alexander von Humboldt Research Award.

Abstract

The classical question in quantitative finance is to provide pricing operators for derivative securities that are in some sense consistent with the observed market prices. Another classical study is to provide hedging strategies. In fact, these two questions are in duality and are always solved together.

Indeed, in the classical theories one assumes an underlying ``historical'' probability measure and all inequalities are understood almost-surely with respect to this measure. In this structure, the classical fundamental theorem of asset pricing (FTAP) states that under the assumption of generalized no-arbitrage, there are risk neutral probability measures which provide linear pricing rules. FTAP proves not only the equivalence between no-arbitrage and the existence of such measures but also shows that these measures are the only possible ones.

In these talks, we consider a more general financial market with Knightian uncertainty. Although in such markets - by definition - there is no historical measure, the basic duality between hedging and pricing still holds. We illustrate this connection by several convex duality results starting with simple discrete time models.